# Symbol-Based Analysis of Finite Element and Isogeometric B-Spline Discretizations of Eigenvalue Problems: Exposition and Review

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## Abstract

We present an example-based exposition and review of recent advances in symbol-based spectral analysis. We consider constant- and variable-coefficient, second-order eigenvalue problems discretized through the (isogeometric) Galerkin method based on B-splines of degree *p* and smoothness \(C^k\), \(0\le k\le p-1\). For each discretized problem, we compute the so-called symbol, which is a function describing the asymptotic singular value and eigenvalue distribution of the associated discretization matrices. Using the symbol, we are able to formulate analytical predictions for the eigenvalue errors occurring when the exact eigenvalues are approximated by the numerical eigenvalues. In this way, we recover and extend previous analytical spectral results. We are also able to predict the existence of \(p-k\) spectral branches, one “acoustical” and \(p-k-1\) “optical”, when discretizing the one-dimensional Laplacian eigenvalue problem. We provide explicit and implicit analytical expressions for these branches, and we quantify the divergence to infinity with respect to *p* of the largest optical branch in the case of \(C^0\) smoothness (the case of classical finite element analysis).

## Notes

### Acknowledgements

The authors express their gratitude to Emanuele Corti for his invaluable informatics support and for his help with the numerical experiments. C. Garoni is a Marie-Curie fellow of INdAM (Istituto Nazionale di Alta Matematica) under grant agreement PCOFUND-GA-2012-600198. H. Speleers was partially supported by the Mission Sustainability Program of the University of Rome Tor Vergata through the project “IDEAS”, and by INdAM Finanziamenti Premiali through the project “SUNRISE”. S-E. Ekström was supported by the FMB – Swedish National Graduate School of Mathematics and Computing. A. Reali was partially supported by Fondazione Cariplo – Regione Lombardia through the project “Verso nuovi strumenti di simulazione super veloci ed accurati basati sull’analisi isogeometrica”, within the program RST – rafforzamento. S. Serra-Capizzano was supported in part by the Donation KAW 2013.0341 from the Knut & Alice Wallenberg Foundation in collaboration with the Royal Swedish Academy of Sciences. T.J.R. Hughes was partially supported by the Office of Naval Research (Grant Nos. N00014-17-1-2119, N00014-17-1-2039, and N00014-13-1-0500), and by the Army Research Office (Grant No. W911NF-13-1-0220). C. Garoni, A. Reali, S. Serra-Capizzano, and H. Speleers are members of the INdAM Research group GNCS (Gruppo Nazionale per il Calcolo Scientifico).

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that there is no conflict of interest.

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